| L(s) = 1 | − 8.27e10·2-s + 5.00e16·3-s + 4.48e21·4-s − 1.31e24·5-s − 4.14e27·6-s − 1.62e30·7-s − 1.76e32·8-s + 2.50e33·9-s + 1.08e35·10-s − 4.64e36·11-s + 2.24e38·12-s + 2.03e39·13-s + 1.34e41·14-s − 6.56e40·15-s + 3.97e42·16-s + 3.84e43·17-s − 2.07e44·18-s + 2.58e45·19-s − 5.88e45·20-s − 8.15e46·21-s + 3.84e47·22-s − 2.47e48·23-s − 8.80e48·24-s − 4.06e49·25-s − 1.68e50·26-s + 1.25e50·27-s − 7.31e51·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.90·4-s − 0.201·5-s − 0.983·6-s − 1.62·7-s − 1.53·8-s + 0.333·9-s + 0.343·10-s − 0.498·11-s + 1.09·12-s + 0.580·13-s + 2.76·14-s − 0.116·15-s + 0.712·16-s + 0.800·17-s − 0.567·18-s + 1.03·19-s − 0.383·20-s − 0.938·21-s + 0.848·22-s − 1.12·23-s − 0.885·24-s − 0.959·25-s − 0.989·26-s + 0.192·27-s − 3.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+71/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(36)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.00e16T \) |
| good | 2 | \( 1 + 8.27e10T + 2.36e21T^{2} \) |
| 5 | \( 1 + 1.31e24T + 4.23e49T^{2} \) |
| 7 | \( 1 + 1.62e30T + 1.00e60T^{2} \) |
| 11 | \( 1 + 4.64e36T + 8.68e73T^{2} \) |
| 13 | \( 1 - 2.03e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 3.84e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 2.58e45T + 6.18e90T^{2} \) |
| 23 | \( 1 + 2.47e48T + 4.81e96T^{2} \) |
| 29 | \( 1 - 8.82e50T + 6.76e103T^{2} \) |
| 31 | \( 1 - 1.59e53T + 7.70e105T^{2} \) |
| 37 | \( 1 + 7.30e55T + 2.19e111T^{2} \) |
| 41 | \( 1 - 3.13e57T + 3.21e114T^{2} \) |
| 43 | \( 1 - 1.90e56T + 9.46e115T^{2} \) |
| 47 | \( 1 + 2.89e59T + 5.23e118T^{2} \) |
| 53 | \( 1 - 2.13e61T + 2.65e122T^{2} \) |
| 59 | \( 1 - 5.88e62T + 5.37e125T^{2} \) |
| 61 | \( 1 - 8.43e62T + 5.73e126T^{2} \) |
| 67 | \( 1 + 5.85e64T + 4.48e129T^{2} \) |
| 71 | \( 1 - 2.84e65T + 2.75e131T^{2} \) |
| 73 | \( 1 + 6.50e65T + 1.97e132T^{2} \) |
| 79 | \( 1 + 2.83e67T + 5.38e134T^{2} \) |
| 83 | \( 1 - 1.64e68T + 1.79e136T^{2} \) |
| 89 | \( 1 - 2.05e69T + 2.55e138T^{2} \) |
| 97 | \( 1 + 5.94e70T + 1.15e141T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.98328812902133120920688678942, −10.20428082434323956989371026446, −9.639289191608152883869433679919, −8.373647451322130763242380848459, −7.35523926064971191895496193846, −6.10993039473384596781949735028, −3.55481566980338597636693528662, −2.50413341950248758511885792562, −1.04567829456314802887228887549, 0,
1.04567829456314802887228887549, 2.50413341950248758511885792562, 3.55481566980338597636693528662, 6.10993039473384596781949735028, 7.35523926064971191895496193846, 8.373647451322130763242380848459, 9.639289191608152883869433679919, 10.20428082434323956989371026446, 11.98328812902133120920688678942
